Paper 2
Instructions to candidates
- Write your session number in the boxes above.
- Do not open this examination paper until instructed to do so.
- A graphic display calculator is required for this paper.
- Section A: answer all questions. Answers must be written within the answer boxes provided.
- Section B: answer all questions in the answer booklet provided. Fill in your session number on the front of the answer booklet, and attach it to this examination paper and your cover sheet using the tag provided.
- Unless otherwise stated in the question, all numerical answers should be given exactly or correct to three significant figures.
- A clean copy of the mathematics HL and further mathematics HL formula booklet is required for this paper.
- The maximum mark for this examination paper is [100 marks].
- Time: 120 minutes
Topic 1 – Core: Algebra – HL Paper 2
- Topic 1.1 :
- Topic 1.2 :
- Topic 1.3 :
- Topic 1.4 :
- Topic 1.5 :
- Complex numbers: the number \({\text{i}} = \sqrt { – 1} \) ; the terms real part, imaginary part, conjugate, modulus and argument.
- Cartesian form \(z = a + {\text{i}}b\) .
- Sums, products and quotients of complex numbers.
- Topic 1.6 :
- Topic 1.7 :
- Topic 1.8 :
- Topic 1.9 :
Topic 2 – Core: Functions and equations
- Topic 2.1
- Topic 2.2
- The graph of a function; its equation \(y = f\left( x \right)\) .
- Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes and symmetry, and consideration of domain and range.
- The graphs of the functions \(y = \left| {f\left( x \right)} \right|\) and \(y = f\left( {\left| x \right|} \right)\)
- The graph of \(\frac{1}{{f\left( x \right)}}\) given the graph of \(y = f(x)\) .
- Topic 2.3
- Topic 2.4
- The rational function \(x \mapsto \frac{{ax + b}}{{cx + d}}\) and its graph.
- The function \(x \mapsto {a^x}\) , \(a > 0\) , and its graph.
- The function \(x \mapsto {\log _a}x\) , \(x > 0\) , and its graph
- Topic 2.5
- Topic 2.6
- Solving quadratic equations using the quadratic formula.
- Use of the discriminant \(\Delta = {b^2} – 4ac\) to determine the nature of the roots.
- Solving polynomial equations both graphically and algebraically.
- Sum and product of the roots of polynomial equations.
- Solution of \({a^x} = b\) using logarithms.
- Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.
- Topic 2.7
Topic 3 – Core: Circular functions and trigonometry
- Topic 3.1 :
- Topic 3.2 :
- Definition of \(\cos \theta \) , \(\sin \theta \) and \(\tan \theta \) in terms of the unit circle.
- Exact values of \(\sin\), \(\cos\) and \(\tan\) of \(0\), \(\frac{\pi }{6}\), \(\frac{\pi }{4}\), \(\frac{\pi }{3}\), \(\frac{\pi }{2}\) and their multiples.
- Definition of the reciprocal trigonometric ratios \(\sec \theta \) , \(\csc \theta \) and \(\cot \theta \) .
- Pythagorean identities: \({\cos ^2}\theta + {\sin ^2}\theta = 1\) ; \(1 + {\tan ^2}\theta = {\sec ^2}\theta \) ; \(1 + {\cot ^2}\theta = {\csc ^2}\theta \) .
- Topic 3.3 :
- Topic 3.4 :
- Topic 3.5 :
- Topic 3.6 :
- Topic 3.7 :
Topic 4 – Core: Vectors
- Topic 4.1
- Concept of a vector.
- Representation of vectors using directed line segments.
- Unit vectors; base vectors \(i\), \(j\), \(k\).
- Components of a vector: \(v = \left( {\begin{array}{*{20}{c}} {{v_1}} \\ {{v_2}} \\ {{v_3}} \end{array}} \right) = {v_1}i + {v_2}j + {v_3}k\) .
- Algebraic and geometric approaches to the sum and difference of two vectors.
- Algebraic and geometric approaches to the zero vector \(0\), the vector \( – v\) .
- Algebraic and geometric approaches to multiplication by a scalar, \(kv\) .
- Algebraic and geometric approaches to magnitude of a vector, \(\left| v \right|\) .
- Algebraic and geometric approaches to position vectors \(\overrightarrow {OA} = a\) .
- \(\overrightarrow {AB} = b – a\) .
- Topic 4.2
- The definition of the scalar product of two vectors.
- Properties of the scalar product: \({\boldsymbol{v}} \cdot {\boldsymbol{w}} = {\boldsymbol{w}} \cdot {\boldsymbol{v}}\) ; \({\boldsymbol{u}} \cdot \left( {{\mathbf{v}} + {\boldsymbol{w}}} \right) = {\boldsymbol{u}} \cdot {\boldsymbol{v}} + {\boldsymbol{u}} \cdot {\boldsymbol{w}}\) ; \(\left( {k{\boldsymbol{v}}} \right) \cdot {\boldsymbol{w}} = k\left( {{\boldsymbol{v}} \cdot {\boldsymbol{w}}} \right)\) ; \({\boldsymbol{v}} \cdot {\boldsymbol{v}} = {\left| {\boldsymbol{v}} \right|^2}\) .
- The angle between two vectors.
- Perpendicular vectors; parallel vectors.
- Topic 4.3
- Topic 4.4
- Topic 4.5
- The definition of the vector product of two vectors.
- Properties of the vector product: \({\text{v}} \times {\text{w}} = – {\text{w}} \times {\text{v}}\) ; \({\text{u}} \times ({\text{v}} + {\text{w}}) = {\text{u}} \times {\text{v}} + {\text{u}} \times {\text{w}}\) ; \((k{\text{v}}) \times {\text{w}} = k({\text{v}} + {\text{w}})\) ; \({\text{v}} \times {\text{v}} = 0\) .
- Geometric interpretation of \({\text{v}} \times {\text{w}}\) .
- Topic 4.6
- Topic 4.7
Topic 5 – Core: Statistics and probability
- Topic 5.1 :
- Topic 5.2 :
- Concepts of trial, outcome, equally likely outcomes, sample space (U) and event.
- The probability of an event \(A\) is \(P\left( A \right) = \frac{{n\left( A \right)}}{{n\left( U \right)}}\)
- The complementary events \(A\) and \({A’}\) (not \(A\)).
- Use of Venn diagrams, tree diagrams, counting principles and tables of outcomes to solve problems.
- Topic 5.3 :
- Topic 5.4 :
- Conditional probability; the definition \(P\left( {\left. A \right|P} \right) = \frac{{P\left( {A\mathop \cap \nolimits B} \right)}}{{P\left( B \right)}}\) .
- Independent events; the definition \(P\left( {\left. A \right|B} \right) = P\left( A \right) = P\left( {\left. A \right|B’} \right)\) .
- Use of Bayes’ theorem for a maximum of three events.
- Topic 5.5 :
- Topic 5.6 :
- Topic 5.7 :
Topic 6 – Core: Calculus
- Topic 6.1
- Informal ideas of limit, continuity and convergence.
- Definition of derivative from first principles as \(f’\left( x \right) = \mathop {\lim }\limits_{h \to 0} {\frac{{f\left( {x + h} \right) – f\left( x \right)}}{h}} \).
- The derivative interpreted as a gradient function and as a rate of change.
- Finding equations of tangents and normals.
- Identifying increasing and decreasing functions.
- The second derivative.
- Higher derivatives.
- Topic 6.2
- Derivatives of \({x^n}\) , \(\sin x\) , \(\cos x\) , \(\tan x\) , \({{\text{e}}^x}\) and \\(\ln x\) .
- Differentiation of sums and multiples of functions.
- The product and quotient rules.
- The chain rule for composite functions.
- Related rates of change.
- Implicit differentiation.
- Derivatives of \(\sec x\) , \(\csc x\) , \(\cot x\) , \({a^x}\) , \({\log _a}x\) , \(\arcsin x\) , \(\arccos x\) and \(\arctan x\) .
- Topic 6.3
- Topic 6.4
- Indefinite integration as anti-differentiation.
- Indefinite integral of \({x^n}\) , \(\sin x\) , \(\cos x\) and \({{\text{e}}^x}\) .
- Other indefinite integrals using the results from 6.2.
- The composites of any of these with a linear function.
- Topic 6.5
- Topic 6.6
- Topic 6.7
Topic 7 – Option: Statistics and probability
- Topic 7.1
- Cumulative distribution functions for both discrete and continuous distributions.
- Geometric distribution.
- Negative binomial distribution.
- Probability generating functions for discrete random variables.
- Using probability generating functions to find mean, variance and the distribution of the sum of \(n\) independent random variables.
- Topic 7.2
- Linear transformation of a single random variable.
- Mean of linear combinations of \(n\) random variables.
- Variance of linear combinations of \(n\) independent random variables.
- Expectation of the product of independent random variables.
- Topic 7.3
- Unbiased estimators and estimates.
- Comparison of unbiased estimators based on variances.
- \({\bar X}\) as an unbiased estimator for \(\mu \) .
- \({S^2}\) as an unbiased estimator for \({\sigma ^2}\) .
- Topic 7.4
- A linear combination of independent normal random variables is normally distributed. In particular, \(X{\text{ ~ }}N\left( {\mu ,{\sigma ^2}} \right) \Rightarrow \bar X{\text{ ~ }}N\left( {\mu ,\frac{{{\sigma ^2}}}{n}} \right)\) .
- The central limit theorem.
- Topic 7.5
- Confidence intervals for the mean of a normal population.
- Topic 7.6
- Null and alternative hypotheses, \({H_0}\) and \({H_1}\) .
- Significance level.
- Critical regions, critical values, \(p\)-values, one-tailed and two-tailed tests.
- Type I and II errors, including calculations of their probabilities.
- Testing hypotheses for the mean of a normal population.
- Topic 7.7
- Introduction to bivariate distributions.
- Covariance and (population) product moment correlation coefficient \(\rho \).
- Proof that \(\rho = 0\) in the case of independence and \( \pm 1\) in the case of a linear relationship between \(X\) and \(Y\).
- Definition of the (sample) product moment correlation coefficient \(R\) in terms of n paired observations on \(X\) and \(Y\).Its application to the estimation of \(\rho \).
- Informal interpretation of \(r\), the observed value of \(R\). Scatter diagrams.
- Topics based on the assumption of bivariate normality: use of the \(t\)-statistic to test the null hypothesis \(\rho = 0\) .
- Topics based on the assumption of bivariate normality: knowledge of the facts that the regression of \(X\) on \(Y\) (\({E\left. {\left( X \right)} \right|Y = y}\)) and \(Y\) on \(X\) (\({E\left. {\left( Y \right)} \right|X = x}\)) are linear.
- Topics based on the assumption of bivariate normality: least-squares estimates of these regression lines (proof not required).
- Topics based on the assumption of bivariate normality: the use of these regression lines to predict the value of one of the variables given the value of the other.
Topic 8 – Option: Sets, relations and groups
- Topic 8.1
- Finite and infinite sets. Subsets.
- Operations on sets: union; intersection; complement; set difference; symmetric difference.
- De Morgan’s laws: distributive, associative and commutative laws (for union and intersection).
- Topic 8.2
- Ordered pairs: the Cartesian product of two sets.
- Relations: equivalence relations; equivalence classes.
- Topic 8.3
- Functions: injections; surjections; bijections.
- Composition of functions and inverse functions.
- Topic 8.4
- Binary operations.
- Operation tables (Cayley tables).
- Topic 8.5
- Binary operations: associative, distributive and commutative properties.
- Topic 8.6
- The identity element \(e\).
- The inverse \({a^{ – 1}}\) of an element \(a\).Proof that left-cancellation and right-cancellation by an element a hold, provided that a has an inverse.
- Proofs of the uniqueness of the identity and inverse elements.
- Topic 8.7
- The definition of a group \(\left\{ {G, * } \right\}\) .
- The operation table of a group is a Latin square, but the converse is false.
- Abelian groups.
- Topic 8.8
- Example of groups: \(\mathbb{R}\), \(\mathbb{Q}\), \(\mathbb{Z}\) and \(\mathbb{C}\) under addition.
- Example of groups: integers under addition modulo \(n\).
- Example of groups: non-zero integers under multiplication, modulo \(p\), where \(p\) is prime.
- Symmetries of plane figures, including equilateral triangles and rectangles.
- Invertible functions under composition of functions.
- Topic 8.9
- The order of a group.
- The order of a group element.
- Cyclic groups.
- Generators.
- Proof that all cyclic groups are Abelian.
- Topic 8.10
- Permutations under composition of permutations.
- Cycle notation for permutations.
- Result that every permutation can be written as a composition of disjoint cycles.
- The order of a combination of cycles.
- Topic 8.11
- Subgroups, proper subgroups.
- Use and proof of subgroup tests.
- Definition and examples of left and right cosets of a subgroup of a group.
- Lagrange’s theorem.
- Use and proof of the result that the order of a finite group is divisible by the order of any element. (Corollary to Lagrange’s theorem.)
- Topic 8.12
- Definition of a group homomorphism.
- Definition of the kernel of a homomorphism.
- Proof that the kernel and range of a homomorphism are subgroups.
- Proof of homomorphism properties for identities and inverses.
- Isomorphism of groups.
- The order of an element is unchanged by an isomorphism.